On Middle Universal Weak and Cross Inverse Property Loops With Equal Length Of Inverse Cycles
|
|
- Sherman Daniels
- 5 years ago
- Views:
Transcription
1 On Middle Universal Weak and Cross Inverse Property Loops With Equal Length Of Inverse Cycles Tèmítópé Gbóláhàn Jaíyéọlá Department of Mathematics, Obafemi Awolowo University, Ile Ife, Nigeria Abstract This study presents a special type of middle isotopism under which the weak inverse property(wip) is isotopic invariant in loops A sufficient condition for a WIPL that is specially isotopic to a loop to be isomorphic to the loop isotope is established Cross inverse property loops(cipls) need not satisfy this sufficient condition It is shown that under this special type of middle isotopism, if n is a positive even integer, then a WIPL has an inverse cycle of length n if and only if its isotope is a WIPL with an inverse cycle of length n But, when n is an odd positive integer If a loop or its isotope is a WIPL with only e and inverse cycles of length n, its isotope or the loop is a WIPL with only e and inverse cycles of length n if and only if they are isomorphic So, that both are isomorphic CIPLs Explanations and procedures are given on how these results can be used to apply CIPLs to cryptography 1 Introduction Let L be a non-empty set Define a binary operation ( ) on L : If x y L for all x, y L, (L, ) is called a groupoid If the system of equations ; a x = b and y a = b have unique solutions for x and y respectively, then (L, ) is called a quasigroup For each x L, the elements x ρ = xj ρ, x λ = xj λ L such that xx ρ = e ρ and x λ x = e λ are called the right, left inverses of x respectively Now, if there exists a unique element e L called the identity element such that for all x L, x e = e x = x, (L, ) is called a loop 2000 Mathematics Subject Classification Primary 20NO5 ; Secondary 08A05 Keywords and Phrases : cross inverse property loops(cipls), weak inverse property loops(wipls), inverse cycles All correspondence to be addressed to this author 1
2 A loop(quasigroup) is a weak inverse property loop(quasigroup)[wipl(wipq)] if and only if it obeys the identity x(yx) ρ = y ρ or (xy) λ x = y λ A loop(quasigroup) is a cross inverse property loop(quasigroup)[cipl(cipq)] if and only if it obeys the identity xy x ρ = y or x yx ρ = y or x λ (yx) = y or x λ y x = y A loop(quasigroup) is an automorphic inverse property loop(quasigroup)[aipl(aipq)] if and only if it obeys the identity (xy) ρ = x ρ y ρ or (xy) λ = x λ y λ Consider (G, ) and (H, ) been two distinct groupoids(quasigroups, loops) Let A, B and C be three distinct non-equal bijective mappings, that maps G onto H The triple α = (A, B, C) is called an isotopism of (G, ) onto (H, ) if and only if xa yb = (x y)c x, y G If α = (A, B, B), then the triple is called a left isotopism and the groupoids(quasigroups, loops) are called left isotopes If α = (A, B, A), then the triple is called a right isotopism and the groupoids(quasigroups, loops) are called right isotopes If α = (A, A, B), then the triple is called a middle isotopism and the groupoids are called middle isotopes If (G, ) = (H, ), then the triple α = (A, B, C) of bijections on (G, ) is called an autotopism of the groupoid(quasigroup, loop) (G, ) Such triples form a group AUT (G, ) called the autotopism group of (G, ) Furthermore, if A = B = C, then A is called an automorphism of the groupoid(quasigroup, loop) (G, ) Such bijections form a group AUM(G, ) called the automorphism group of (G, ) As observed by Osborn [16], a loop is a WIPL and an AIPL if and only if it is a CIPL The past efforts of Artzy [1, 4, 3, 2], Belousov and Tzurkan [5] and present studies of Keedwell [10], Keedwell and Shcherbacov [11, 12, 13] are of great significance in the study of WIPLs, AIPLs, CIPQs and CIPLs, their generalizations(ie m-inverse loops and quasigroups, (r,s,t)- inverse quasigroups) and applications to cryptography In the quest for the application of CIPQs with long inverse cycles to cryptography, Keedwell [10] constructed a CIPQ The author also gave examples and detailed explanation and procedures of the use of this CIPQ for cryptography Cross inverse property quasigroups have been found appropriate for cryptography because of the fact that the left and right inverses x λ and x ρ of an element x do not coincide unlike in left and right inverse property loops, hence this gave rise to what is called cycle of inverses or inverse cycles or simply cycles ie finite sequence of elements x 1, x 2,, x n such that x ρ k = x k+1 mod n The 2
3 number n is called the length of the cycle The origin of the idea of cycles can be traced back to Artzy [1, 4] where he also found there existence in WIPLs apart form CIPLs In his two papers, he proved some results on possibilities for the values of n and for the number m of cycles of length n for WIPLs and especially CIPLs We call these Cycle Theorems for now The universality of WIPLs and CIPLs have been addressed by OSborn [16] and Artzy [2] respectively Artzy showed that isotopic CIPLs are isomorphic In 1970, Basarab [7] later continued the work of Osborn of 1961 on universal WIPLs by studying isotopes of WIPLs that are also WIPLs after he had studied a class of WIPLs([6]) in 1967 Osborn [16], while investigating the universality of WIPLs discovered that a universal WIPL (G, ) obeys the identity yx (ze y y) = (y xz) y x, y, z G (1) where E y = L y L y λ = R 1 y ρ R 1 y = L y R y L 1 y Ry 1 Eight years after Osborn s [16] 1960 work on WIPL, in 1968, Huthnance Jr [9] studied the theory of generalized Moufang loops He named a loop that obeys (1) a generalized Moufang loop and later on in the same thesis, he called them M-loops On the other hand, he called a universal WIPL an Osborn loop and this same definition was adopted by Chiboka [8] From the literature review stated above, it can be seen that neither WIPLs nor CIPLs has been shown to be isotopic invariant In fact, it is yet to be shown that there exist a special type of isotopism(eg left, right or middle isotopism) under which the WIP or CIP is isotopic invariant The aim of the present study is to present a special type of middle isotopism under which the WIP is isotopic invariant in loops A sufficient condition for a WIPL that is specially isotopic to a loop to be isomorphic to the loop isotope is established Cross inverse property loops(cipls) need not satisfy this sufficient condition It is shown that under this special type of middle isotopism, if n is a positive even integer, then a WIPL has an inverse cycle of length n if and only if its isotope is a WIPL with an inverse cycle of length n But, when n is an odd positive integer If a loop or its isotope is a WIPL with only e and inverse cycles of length n, its isotope or the loop is a WIPL with only e and inverse cycles of length n if and only if they are isomorphic So, that both are isomorphic CIPLs Explanations and procedures are given on how these results can be used to apply CIPLs to cryptography 2 Preliminaries Definition 21 Let L be a loop A mapping α SY M(L)(where SY M(L) is the group of all bijections on L) which obeys the identity x ρ = [(xα) ρ ]α is called a weak right inverse permutation Their set is represented by S ρ (L) Similarly, if α obeys the identity x λ = [(xα) λ ]α it is called a weak left inverse permutation Their set is represented by S λ (L) If α satisfies both, it is called a weak inverse permutation Their set is represented by S (L) 3
4 It can be shown that α S(L) is a weak right inverse if and only if it is a weak left inverse permutation So, S (L) = S ρ (L) = S λ (L) Remark 21 Every permutation of order 2 that preserves the right(left) inverse of each element in a loop is a weak right (left) inverse permutation Example 21 If L is an extra loop, the left and right inner mappings L(x, y) and R(x, y) x, y L are automorphisms of orders 2 ([14]) Hence, they are weak inverse permutations by Remark 21 Throughout, we shall employ the use of the bijections; J ρ : x x ρ, J λ : x x λ, L x : y xy and R x : y yx for a loop and the bijections; J ρ : x x ρ, J λ : x xλ, L x : y xy and R x : y yx for its loop isotope If the identity element of a loop is e then that of the isotope shall be denoted by e Lemma 21 In a loop, the set of weak inverse permutations that commute form an abelian group Remark 22 Applying Lemma 21 to extra loops and considering Example 21, it will be observed that in an extra loop L, the Boolean groups Inn λ (L), Inn ρ S (L) Inn λ (L) and Inn ρ (L) are the left and right inner mapping groups respectively They have been investigated in [15] and [14] This deductions can t be drawn for CC-loops despite the fact that the left (right) inner mappings commute and are automorphisms And this is as a result of the fact that the left(right) inner mappings are not of exponent 2 Definition 22 (T -condition) Let (G, ) and (H, ) be two distinct loops that are isotopic under the triple (A, B, C) (G, ) obeys the T 1 condition if and only if A = B (G, ) obeys the T 2 condition if and only if J ρ = C 1 J ρ B = A 1 J ρ C (G, ) obeys the T 3 condition if and only if J λ = C 1 J λ A = B 1 J λ C So, (G, ) obeys the T condition if and only if it obey T 1 and T 2 conditions or T 1 and T 3 conditions since T 2 T 3 It must here by be noted that the T -conditions refer to a pair of isotopic loops at a time This statement might be omitted at times That is whenever we say a loop (G, ) has the T -condition, then this is relative to some isotope (H, ) of (G, ) Lemma 22 Let L be a loop The following are equivalent 1 L is a WIPL 2 R y J ρ L y = J ρ y L 3 L x J λ R x = J λ x L Lemma 23 (Lemma, Artzy [4]) Let a WIPL consist only of e and inverse cycles of length n If n is odd, the loop is a CIPL 4
5 3 Main Results Theorem 31 Let (G, ) and (H, ) be two distinct loops that are isotopic under the triple (A, B, C) 1 If the pair of (G, ) and (H, ) obey the T condition, then (G, ) is a WIPL if and only if (H, ) is a WIPL 2 If (G, ) and (H, ) are WIPLs, then J λ R x J ρ B λ R xa J ρ and J ρ L x J λ A ρl xb J λ for all x G 1 (A, B, C) : G H is an isotopism xa yb = (x y)c ybl xa = yl xc BL xa = L xc L xa = B 1 L x C L x = BL xac 1 (2) Also, (A, B, C) : G H is an isotopism xar yb = xr yc AR yb = R yc R yb = A 1 R y C R y = AR ybc 1 (3) Applying (2) and (3) to Lemma 22 separately, we have : R y J ρ L y = J ρ, L x J λ R x = J λ (AR xb C 1 )J ρ (BL xa C 1 ) = J ρ, (BL xa C 1 )J λ (AR xb C 1 ) = J λ AR xb (C 1 J ρ B)L xa C 1 = J ρ, BL xa (C 1 J λ A)R xb C 1 = J λ R xb(c 1 J ρ B)L xa = A 1 J ρ C, L xa(c 1 J λ A)R xb = B 1 J λ C (4) Let J ρ = C 1 J ρ B = A 1 J ρ C, J λ = C 1 J λ A = B 1 J λ C Then, from (4) and by Lemma 22, H is a WIPL if xb = xa and J ρ = C 1 J ρ B = A 1 J ρ C or xa = xb and J λ = C 1 J λ A = B 1 J λ C B = A and J ρ = C 1 J ρ B = A 1 J ρ C or A = B and J λ = C 1 J λ A = B 1 J λ C A = B and J ρ = C 1 J ρ B = A 1 J ρ C or J λ = C 1 J λ A = B 1 J λ C This completes the proof of the forward part To prove the converse, carry out the same procedure, assuming the T condition and the fact that (H, ) is a WIPL 2 If (H, ) is a WIPL, then while since G is a WIPL, The fact that G and H are isotopic implies that R yj ρl y = J ρ, y H (5) R x J ρ L x = J ρ x G (6) L x = BL xac 1 x G and (7) R x = AR xbc 1 x G (8) 5
6 From (5), while from (6), So, using (10) and (12) in (7) we get while using (9) and (11) in (8) we get R y = J ρl 1 y J λ y H and (9) L y = J λr 1 y J ρ y H (10) R x = J ρ L 1 x J λ x G and (11) L x = J λ R 1 x J ρ x G (12) J λ R x J ρ B λr xaj ρ x G (13) J ρ L x J λ A ρl xbj λ x G (14) Theorem 32 Let (G, ) and (H, ) be two distinct loops that are isotopic under the triple (A, B, C) such that they obey the T condition 1 If n is an even positive integer Then, (G, ) is a WIPL with an inverse cycle of length n if and only if (H, ) is a WIPL with an inverse cycle of length n 2 When n is an odd positive integer If (G, ) or (H, ) is a WIPL with only e and inverse cycles of length n, (H, ) or (G, ) is a WIPL with only e and inverse cycles of length n if and only if (G, ) = (H, ) So, (G, ) and (H, ) are isomorphic CIPLs The fact that (G, ) is a WIPL if and only if (H, ) is a WIPL has been proved in Theorem 31 1 It will now be shown that (G, ) has an inverse cycle of length n if and only if (H, ) is a WIPL with an inverse cycle of length n A WIPL (G, ) has an inverse cycle of length n if and only if J ρ = n Recall that J ρ ρb 1 and J ρ = AJ ρc 1 Consider the following inductive process J m ρ Jρ 2 ρb 1 AJ ρc 1 2 2=2 1, Jρ 4 ρ 2 C 1 CJ 2 } {{ } 4=2 2 Jρ 6 ρ 4 C 1 CJ ρ 2 C 1 ρ 6 C 1, Jρ 8 ρ 6 C 1 CJ ρ 2 C 1 6=2 3 8=2 4 ρ m 2 C 1 CJ ρ 2 C 1 m=2 k m, J m+2 ρ ρ m C 1 CJ ρ 2 C 1 m+2=2 (k+1) 4 8 ρ m+2 C 1 Jρ n ρ n 2 C 1 CJ 2 n=2 (m+1) n, J n+2 ρ ρ n C 1 CJ 2 } {{ } n+2=2 (m+2) n+2 6
7 So, J n ρ n for all even n Z + Thus, J ρ = n if and only if J ρ = n which justifies the claim 2 Let n be an odd positive integer and consider the following inductive process J m ρ Jρ 2 ρb 1 AJ ρc 1 ρ 2 C 1, Jρ 3 ρ 2 C 1 CJ ρb 1 2=2 1 Jρ 4 ρ 2 C 1 CJ ρ 2 C 1 ρ 4 C 1, Jρ 5 ρ 4 C 1 CJ ρb 1 4=2 2 ρ m 2 C 1 CJ 2 m, J m+1 ρ ρ m C 1 CJ ρb 1 3 ρ B 1 5 ρ B 1 m+1 ρ B 1 Jρ n ρ n 2 C 1 CJ ρ 2 C 1 n, J n+1 ρ ρ n C 1 CJ ρb 1 n+1 ρ B 1 So, J n ρ n for all odd n Z + Thus, if J ρ = n or J ρ = n then, J ρ = n or J ρ = n if and only if (G, ) = (H, ) which justifies the claim By Lemma 23, G and H are CIPLs Corollary 31 Let (G, ) and (H, ) be two distinct loops that are isotopic under the triple (A, B, C) If G is a WIPL with the T condition, then H is a WIPL and so: 1 there exists α, β S (G) ie α and β are weak inverse permutations and 2 J ρ = J λ J ρ = J λ By Theorem 31, A = B and J ρ = C 1 J ρ B = A 1 J ρ C or J λ = C 1 J λ A = B 1 J λ C 1 C 1 J ρ B = A 1 J ρ C J ρ B = CA 1 J ρ C J ρ = CA 1 J ρ CB 1 = CA 1 J ρ CA 1 = αj ρ α where α = CA 1 S(G, ) 2 C 1 J λ A = B 1 J λ C J λ A = CB 1 J λ C J λ = CB 1 J λ CA 1 = CB 1 J λ CB 1 = βj λ β where β = CB 1 S(G, ) 3 J ρ = C 1 J ρ B, J λ = C 1 J λ A J ρ = J λ C 1 J ρ B = C 1 J λ A = C 1 J λ B J λ = J ρ Lemma 31 Let (G, ) be a WIPL with the T condition and isotopic to another loop (H, ) (H, ) is a WIPL and G has a weak inverse permutation From the proof of Corollary 31, α = β, hence the conclusion 7
8 Theorem 33 If two distinct loops are isotopic under the T condition And any one of them is a WIPL and has a trivial set of weak inverse permutations, then the two loops are both WIPLs that are isomorphic From Lemma 31, α = I is a weak inverse permutation In the proof of Corollary 31, α = CA 1 = I A = C Already, A = B, hence (G, ) = (H, ) Remark 31 Theorem 33 describes isotopic WIP loops that are isomorphic by the T condition(for a special case) Application To Cryptography In application, it is assumed that the message to be transmitted can be represented as single element x of a loop (G, ) and that this is enciphered by multiplying by another element y of G so that the encoded message is yx At the receiving end, the message is deciphered by multiplying by the right inverse y ρ of y Let (G, ) be a WIPL with only e and inverse cycles of length n where n is an odd positive integer So it is a CIPL Let (H, ) be a loop that is isotopic to (G, ) under the T condition such that (G, ) = (H, ) Then by Theorem 31, H is a WIPL but by Theorem 32, H does not have only e and inverse cycles of length n and so it is not a CIPL So, according to Theorem 31, by the choice of the triple (A, B, C) been an isotopism from G onto H such that the T condition holds, if G is a CIPL then H is a WIPL that is not a CIPL So, the secret key for the systems is the pair {(A, B, C), T } Thus whenever a set of information or messages is to be transmitted, the sender will enciphere in G(as described earlier on) and then enciphere again with {(A, B, C), T } to get a WIPL H which is the set of encoded messages At the receiving end, the combined message H is deciphered by using an inverse isotopism(ie inverse key {(A, B 1, C 1 ), T }) to get G and then deciphere again(as described earlier on) to get the messages The secret key can be changed over time The method described above is a double encryption and its a double protection It protects each piece of information(element of the loop) and protects the combined information(the loop as a whole) Its like putting on a pair of socks and shoes or putting on under wears and clothes, the body gets better protection 4 Conclusion and Future Study Karklinüsh and Karkliň [11] introduced m-inverse loops ie loops that obey any of the equivalent conditions (xy)j m ρ xj m+1 ρ = yj m ρ and xj m+1 λ (yx)j m λ = yj m λ They are generalizations of WIPLs and CIPLs, which corresponds to m = 1 and m = 0 respectively After the study of m-loops by Keedwell and Shcherbacov [10], they have also generalized them to quasigroups called (r, s, t)-inverse quasigroups in [12] and [13] It will be interesting to study the universality of m-inverse loops and (r, s, t)-inverse quasigroups These will generalize the works of J M Osborn and R Artzy on universal WIPLs and CIPLs respectively 8
9 References [1] R Artzy (1955), On loops with special property, Proc Amer Math Soc 6, [2] R Artzy (1959), Crossed inverse and related loops, Trans Amer Math Soc 91, 3, [3] R Artzy (1959), On Automorphic-Inverse Properties in Loops, Proc Amer Math Soc 10,4, [4] R Artzy (1978), Inverse-Cycles in Weak-Inverse Loops, Proc Amer Math Soc 68, 2, [5] V D Belousov (1969), Crossed inverse quasigroups(ci-quasigroups), Izv Vyss Ucebn; Zaved Matematika 82, [6] A S Basarab (1967), A class of WIP-loops, Mat Issled 2(2), 3-24 [7] A S Basarab (1970), Isotopy of WIP loops, Mat Issled 5, 2(16), 3-12 [8] V O Chiboka (1990), The study of properties and construction of certain finite order G-loops, PhD thesis, Obafemi Awolowo University, Ile-Ife [9] E D Huthnance Jr(1968), A theory of generalised Moufang loops, PhD thesis, Georgia Institute of Technology [10] A D Keedwell (1999), Crossed-inverse quasigroups with long inverse cycles and applications to cryptography, Australas J Combin 20, [11] A D Keedwell and V A Shcherbacov (2002), On m-inverse loops and quasigroups with a long inverse cycle, Australas J Combin 26, [12] A D Keedwell and V A Shcherbacov (2003), Construction and properties of (r, s, t)- inverse quasigroups I, Discrete Math 266, [13] A D Keedwell and V A Shcherbacov, Construction and properties of (r, s, t)-inverse quasigroups II, Discrete Math 288 (2004), [14] M K Kinyon, K Kunen (2004), The structure of extra loops, Quasigroups and Related Systems 12, [15] M K Kinyon, K Kunen, J D Phillips (2004), Diassociativity in conjugacy closed loops, Comm Alg 32, [16] J M Osborn (1961), Loops with the weak inverse property, Pac J Math 10,
On Middle Universal m-inverse Quasigroups And Their Applications To Cryptography
On Middle Universal m-inverse Quasigroups And Their Applications To Cryptography Tèmítópé Gbóláhàn Jaíyéọlá Department of Mathematics, Obafemi Awolowo University, Ile Ife, Nigeria jaiyeolatemitope@yahoocom,
More informationSome isotopy-isomorphy conditions for m-inverse quasigroups and loops
An. Şt. Univ. Ovidius Constanţa Vol. 16(2), 2008, 57 66 Some isotopy-isomorphy conditions for m-inverse quasigroups and loops Tèmító. pé. Gbó. láhàn JAÍYÉỌLÁ Abstract This work presents a special type
More informationAn holomorphic study of Smarandache automorphic and cross inverse property loops
Scientia Magna Vol. 4 (2008), No. 1, 102-108 An holomorphic study of Smarandache automorphic and cross inverse property loops Tèmító. pé. Gbó. láhàn Jaíyéọlá Department of Mathematics, Obafemi Awolowo
More informationOn A Cryptographic Identity In Osborn Loops
On A Cryptographic Identity In Osborn Loops T. G. Jaiyéọlá Department of Mathematics, Obafemi Awolowo University, Ile Ife 220005, Nigeria. jaiyeolatemitope@yahoo.com tjayeola@oauife.edu.ng J. O. Adéníran
More informationA Pair of Smarandachely Isotopic Quasigroups and Loops of the Same Variety
International J.Math. Combin. Vol.1 (2008), 36-44 A Pair of Smarandachely Isotopic Quasigroups and Loops of the Same Variety Tèmító. pé. Gbó. láhàn Jaíyéọlá (Department of Mathematics, Obafemi Awolowo
More informationON THE UNIVERSALITY OF SOME SMARANDACHE LOOPS OF BOL-MOUFANG TYPE
arxiv:math/0607739v2 [math.gm] 8 Sep 2007 ON THE UNIVERSALITY OF SOME SMARANDACHE LOOPS OF BOL-MOUFANG TYPE Tèmítópé Gbóláhàn Jaíyéọlá Department o Mathematics, Obaemi Awolowo University, Ile Ie, Nieria.
More informationOn central loops and the central square property
On central loops and the central square property arxiv:0707.1441v [math.gm] 5 Jun 008 Tèmító.pé. Gbó.láhàn Jaiyéọlá 1 & John Olúsọlá Adéníran Abstract The representation sets of a central square C-loop
More informationBasic Properties Of Second Smarandache Bol Loops
Basic Properties Of Second Smarandache Bol Loops Tèmító.pé. Gbó.láhàn Jaíyéọlá Department of Mathematics, Obafemi Awolowo University, Ile Ife, Nigeria. jaiyeolatemitope@yahoo.com, tjayeola@oauife.edu.ng
More informationCheban loops. 1 Introduction
Ashdin Publishing Journal of Generalized Lie Theory and Applications Vol. 4 (2010), Article ID G100501, 5 pages doi:10.4303/jglta/g100501 Cheban loops J. D. Phillips a and V. A. Shcherbacov b a Department
More informationHALF-ISOMORPHISMS OF MOUFANG LOOPS
HALF-ISOMORPHISMS OF MOUFANG LOOPS MICHAEL KINYON, IZABELLA STUHL, AND PETR VOJTĚCHOVSKÝ Abstract. We prove that if the squaring map in the factor loop of a Moufang loop Q over its nucleus is surjective,
More informationA CLASS OF LOOPS CATEGORICALLY ISOMORPHIC TO UNIQUELY 2-DIVISIBLE BRUCK LOOPS
A CLASS OF LOOPS CATEGORICALLY ISOMORPHIC TO UNIQUELY 2-DIVISIBLE BRUCK LOOPS MARK GREER Abstract. We define a new variety of loops we call Γ-loops. After showing Γ-loops are power associative, our main
More informationOn m-inverse loops and quasigroups with a long inverse cycle
On m-inverse loops and quasigroups with a long inverse cycle A. D. Keedwell Department of Mathematics and Statistics University of Surrey Guildford GU2 7XH Surrey, United Kingdom V. A. Shcherbacov Institute
More informationOn Mikheev s construction of enveloping groups
Comment.Math.Univ.Carolin. 51,2(2010) 245 252 245 On Mikheev s construction of enveloping groups J.I. Hall Astract. Mikheev, starting from a Moufang loop, constructed a groupoid and reported that this
More informationNew Algebraic Properties of Middle Bol Loops
New Algebraic Properties of Middle Bol Loops arxiv:1606.09169v1 [math.gr] 28 Jun 2016 T. G. Jaiyéọlá Department of Mathematics, Obafemi Awolowo University, Ile Ife 220005, Nigeria. jaiyeolatemitope@yahoo.com
More informationNILPOTENCY IN AUTOMORPHIC LOOPS OF PRIME POWER ORDER
NILPOTENCY IN AUTOMORPHIC LOOPS OF PRIME POWER ORDER PŘEMYSL JEDLIČKA, MICHAEL KINYON, AND PETR VOJTĚCHOVSKÝ Abstract. A loop is automorphic if its inner mappings are automorphisms. Using socalled associated
More informationConjugate Loops: An Introduction
International Mathematical Forum, Vol. 8, 2013, no. 5, 223-228 Conjugate Loops: An Introduction A. Batool, A. Shaheen and A. Ali Department of Mathematics Quaid-i-Azam University, Islamabad, Pakistan afshan_batoolqau@yahoo.com
More informationKEN KUNEN: ALGEBRAIST
KEN KUNEN: ALGEBRAIST MICHAEL KINYON 1. Introduction Ken Kunen is justifiably best known for his work in set theory and topology. What I would guess many of his friends and students in those areas do not
More informationTHE STRUCTURE OF AUTOMORPHIC LOOPS
THE STRUCTURE OF AUTOMORPHIC LOOPS MICHAEL K. KINYON, KENNETH KUNEN, J. D. PHILLIPS, AND PETR VOJTĚCHOVSKÝ Abstract. Automorphic loops are loops in which all inner mappings are automorphisms. This variety
More informationCrossed-inverse quasigroups with long inverse cycles and applications to cryptography
Crossed-inverse quasigroups with long inverse cycles and applications to cryptography A. D. Keedwell Department of Mathematics and Statistics University of Surrey, Guildford GU2 5XH, Surrey, U.K. Abstract
More informationOctonions. Robert A. Wilson. 24/11/08, QMUL, Pure Mathematics Seminar
Octonions Robert A. Wilson 4//08, QMUL, Pure Mathematics Seminar Introduction This is the second talk in a projected series of five. I shall try to make them as independent as possible, so that it will
More informationGroups and Symmetries
Groups and Symmetries Definition: Symmetry A symmetry of a shape is a rigid motion that takes vertices to vertices, edges to edges. Note: A rigid motion preserves angles and distances. Definition: Group
More informationSemigroup, monoid and group models of groupoid identities. 1. Introduction
Quasigroups and Related Systems 16 (2008), 25 29 Semigroup, monoid and group models of groupoid identities Nick C. Fiala Abstract In this note, we characterize those groupoid identities that have a (nite)
More informationAutomorphism Groups of Simple Moufang Loops over Perfect Fields
Automorphism Groups of Simple Moufang Loops over Perfect Fields By GÁBOR P. NAGY SZTE Bolyai Institute Aradi vértanúk tere 1, H-6720 Szeged, Hungary e-mail: nagyg@math.u-szeged.hu PETR VOJTĚCHOVSKÝ Department
More informationMath 120: Homework 6 Solutions
Math 120: Homewor 6 Solutions November 18, 2018 Problem 4.4 # 2. Prove that if G is an abelian group of order pq, where p and q are distinct primes then G is cyclic. Solution. By Cauchy s theorem, G has
More informationREVERSALS ON SFT S. 1. Introduction and preliminaries
Trends in Mathematics Information Center for Mathematical Sciences Volume 7, Number 2, December, 2004, Pages 119 125 REVERSALS ON SFT S JUNGSEOB LEE Abstract. Reversals of topological dynamical systems
More informationAUTOMORPHISMS OF DIHEDRAL-LIKE AUTOMORPHIC LOOPS
AUTOMORPHISMS OF DIHEDRAL-LIKE AUTOMORPHIC LOOPS MOUNA ABORAS AND PETR VOJTĚCHOVSKÝ Abstract. Automorphic loops are loops in which all inner mappings are automorphisms. A large class of automorphic loops
More informationAutomated theorem proving in algebra
Automated theorem proving in algebra David Stanovský Charles University in Prague Czech Republic stanovsk@karlin.mff.cuni.cz http://www.karlin.mff.cuni.cz/ stanovsk Beograd, November 2009 David Stanovský
More informationCommentationes Mathematicae Universitatis Carolinae
Commentationes Mathematicae Universitatis Carolinae Mark Greer; Lee Raney Moufang semidirect products of loops with groups and inverse property extensions Commentationes Mathematicae Universitatis Carolinae,
More informationNormal Automorphisms of Free Burnside Groups of Period 3
Armenian Journal of Mathematics Volume 9, Number, 017, 60 67 Normal Automorphisms of Free Burnside Groups of Period 3 V. S. Atabekyan, H. T. Aslanyan and A. E. Grigoryan Abstract. If any normal subgroup
More informationarxiv: v2 [math.rt] 3 Mar 2014
Moufang symmetry I Generalized Lie and Maurer-Cartan equations Eugen Paal arxiv:08023471v2 [mathrt] 3 Mar 2014 Abstract The continuous Moufang loops are characterized as the algebraic systems where the
More informationON THE ORDER OF ARC-STABILISERS IN ARC-TRANSITIVE GRAPHS, II
Bull. Aust. Math. Soc. 87 (2013), 441 447 doi:10.1017/s0004972712000470 ON THE ORDER OF ARC-STABILISERS IN ARC-TRANSITIVE GRAPHS, II GABRIEL VERRET (Received 30 April 2012; accepted 11 May 2012; first
More informationLoops of Bol-Moufang type with a subgroup of index two
BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Number 2(45), 2004, Pages 1 17 ISSN 1024 7696 Loops of Bol-Moufang type with a subgroup of index two M. K. Kinyon, J. D. Phillips, and P.
More informationLOOPS OF BOL-MOUFANG TYPE WITH A SUBGROUP OF INDEX TWO
LOOPS OF BOL-MOUFANG TYPE WITH A SUBGROUP OF INDEX TWO MICHAEL K. KINYON, J. D. PHILLIPS, AND PETR VOJTĚCHOVSKÝ Abstract. We describe all constructions for loops of Bol-Moufang type analogous to the Chein
More informationRealizations of Loops and Groups defined by short identities
Comment.Math.Univ.Carolin. 50,3(2009) 373 383 373 Realizations of Loops and Groups defined by short identities A.D. Keedwell Abstract. In a recent paper, those quasigroup identities involving at most three
More informationTHE GROUP OF UNITS OF SOME FINITE LOCAL RINGS I
J Korean Math Soc 46 (009), No, pp 95 311 THE GROUP OF UNITS OF SOME FINITE LOCAL RINGS I Sung Sik Woo Abstract The purpose of this paper is to identify the group of units of finite local rings of the
More informationSmarandache isotopy of second Smarandache Bol loops
Scientia Mana Vol. 7 (2011), No. 1, 82-93 Smarandache isotopy o second Smarandache Bol loops Tèmító. pé. Gbó. láhàn Jaíyéọlá Department o Mathematics, Obaemi Awolowo University, Ile Ie 220005, Nieria E-mail:
More informationSYLOW THEORY FOR QUASIGROUPS II: SECTIONAL ACTION
SYLOW THEORY FOR QUASIGROUPS II: SECTIONAL ACTION MICHAEL K. KINYON 1, JONATHAN D. H. SMITH 2, AND PETR VOJTĚCHOVSKÝ3 Abstract. The first paper in this series initiated a study of Sylow theory for quasigroups
More informationOn certain Regular Maps with Automorphism group PSL(2, p) Martin Downs
BULLETIN OF THE GREEK MATHEMATICAL SOCIETY Volume 53, 2007 (59 67) On certain Regular Maps with Automorphism group PSL(2, p) Martin Downs Received 18/04/2007 Accepted 03/10/2007 Abstract Let p be any prime
More informationTwo questions on semigroup laws
Two questions on semigroup laws O. Macedońska August 17, 2013 Abstract B. H. Neumann recently proved some implication for semigroup laws in groups. This may help in solution of a problem posed by G. M.
More informationOn the algebraic structure of Dyson s Circular Ensembles. Jonathan D.H. SMITH. January 29, 2012
Scientiae Mathematicae Japonicae 101 On the algebraic structure of Dyson s Circular Ensembles Jonathan D.H. SMITH January 29, 2012 Abstract. Dyson s Circular Orthogonal, Unitary, and Symplectic Ensembles
More informationMAXIMAL SUBALGEBRAS AND CHIEF FACTORS OF LIE ALGEBRAS DAVID A. TOWERS
MAXIMAL SUBALGEBRAS AND CHIEF FACTORS OF LIE ALGEBRAS DAVID A. TOWERS Department of Mathematics and Statistics Lancaster University Lancaster LA1 4YF England d.towers@lancaster.ac.uk Abstract This paper
More informationPALINDROMIC AND SŪDOKU QUASIGROUPS
PALINDROMIC AND SŪDOKU QUASIGROUPS JONATHAN D. H. SMITH Abstract. Two quasigroup identities of importance in combinatorics, Schroeder s Second Law and Stein s Third Law, share many common features that
More informationRohit Garg Roll no Dr. Deepak Gumber
FINITE -GROUPS IN WHICH EACH CENTRAL AUTOMORPHISM FIXES THE CENTER ELEMENTWISE Thesis submitted in partial fulfillment of the requirement for the award of the degree of Masters of Science In Mathematics
More informationInverse Properties in Neutrosophic Triplet Loop and Their Application to Cryptography
algorithms Article Inverse Properties in Neutrosophic Triplet Loop and Their Application to Cryptography Temitope Gbolahan Jaiyeola 1 ID and Florentin Smarandache 2, * ID 1 Department of Mathematics, Obafemi
More informationThe upper triangular algebra loop of degree 4
The upper triangular algebra loop of degree 4 Michael Munywoki Joint With K.W. Johnson and J.D.H. Smith Iowa State University Department of Mathematics Third Mile High Conference on Nonassociative Mathematics
More informationON VARIETIES OF LEFT DISTRIBUTIVE LEFT IDEMPOTENT GROUPOIDS
ON VARIETIES OF LEFT DISTRIBUTIVE LEFT IDEMPOTENT GROUPOIDS DAVID STANOVSKÝ Abstract. We describe a part of the lattice of subvarieties of left distributive left idempotent groupoids (i.e. those satisfying
More informationSEMIGROUPS OF TRANSFORMATIONS WITH INVARIANT SET
J Korean Math Soc 48 (2011) No 2 pp 289 300 DOI 104134/JKMS2011482289 SEMIGROUPS OF TRANSFORMATIONS WITH INVARIANT SET Preeyanuch Honyam and Jintana Sanwong Abstract Let T (X) denote the semigroup (under
More information6 Permutations Very little of this section comes from PJE.
6 Permutations Very little of this section comes from PJE Definition A permutation (p147 of a set A is a bijection ρ : A A Notation If A = {a b c } and ρ is a permutation on A we can express the action
More informationON STRONGLY PRIME IDEALS AND STRONGLY ZERO-DIMENSIONAL RINGS. Christian Gottlieb
ON STRONGLY PRIME IDEALS AND STRONGLY ZERO-DIMENSIONAL RINGS Christian Gottlieb Department of Mathematics, University of Stockholm SE-106 91 Stockholm, Sweden gottlieb@math.su.se Abstract A prime ideal
More informationORDERED POWER ASSOCIATIVE GROUPOIDS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 31, No. 1, January 1972 ORDERED POWER ASSOCIATIVE GROUPOIDS DESMOND A. ROBBIE Abstract. Compact, connected, totally ordered, (Hausdorff) topological
More informationarxiv: v3 [math.gr] 23 Feb 2010
THE STRUCTURE OF COMMUTATIVE AUTOMORPHIC LOOPS arxiv:0810.1065v3 [math.gr] 23 Feb 2010 Abstract. An automorphic loop (or A-loop) is a loop whose inner mappings are automorphisms. Every element of a commutative
More information6 Orthogonal groups. O 2m 1 q. q 2i 1 q 2i. 1 i 1. 1 q 2i 2. O 2m q. q m m 1. 1 q 2i 1 i 1. 1 q 2i. i 1. 2 q 1 q i 1 q i 1. m 1.
6 Orthogonal groups We now turn to the orthogonal groups. These are more difficult, for two related reasons. First, it is not always true that the group of isometries with determinant 1 is equal to its
More informationNAPOLEON S QUASIGROUPS
NAPOLEON S QUASIGROUPS VEDRAN KRČADINAC Abstract. Napoleon s quasigroups are idempotent medial quasigroups satisfying the identity (ab b)(b ba) = b. In works by V.Volenec geometric terminology has been
More informationClassifying Bol-Moufang Quasigroups Under a Single Operation
Classifying Bol-Moufang Quasigroups Under a Single Operation Ashley Arp Jasmine Van Exel Michael Kaminski Davian Vernon Black Hills State University Paine College Knox College Morehouse College Abstract
More informationOn orthogonality of binary operations and squares
BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Number 2(48), 2005, Pages 3 42 ISSN 1024 7696 On orthogonality of binary operations and squares Gary L. Mullen, Victor A. Shcherbacov Abstract.
More informationA GRAPH-THEORETIC APPROACH TO QUASIGROUP CYCLE NUMBERS
A GRAPH-THEORETIC APPROACH TO QUASIGROUP CYCLE NUMBERS BRENT KERBY AND JONATHAN D. H. SMITH Abstract. Norton and Stein associated a number with each idempotent quasigroup or diagonalized Latin square of
More informationRecognising nilpotent groups
Recognising nilpotent groups A. R. Camina and R. D. Camina School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, UK; a.camina@uea.ac.uk Fitzwilliam College, Cambridge, CB3 0DG, UK; R.D.Camina@dpmms.cam.ac.uk
More informationBY MolmS NEWMAN (1011) (11
THE STRUCTURE OF SOME SUBGROUPS OF THE MODULAR GROUP BY MolmS NEWMAN Introduction Let I be the 2 X 2 modular group. In a recent article [7] the notion of the type of a subgroup A of r was introduced. If
More informationSome results on primeness in the near-ring of Lipschitz functions on a normed vector space
Hacettepe Journal of Mathematics and Statistics Volume 43 (5) (014), 747 753 Some results on primeness in the near-ring of Lipschitz functions on a normed vector space Mark Farag Received 01 : 0 : 013
More informationarxiv: v1 [math.gt] 11 Aug 2008
Link invariants from finite Coxeter racks Sam Nelson Ryan Wieghard arxiv:0808.1584v1 [math.gt] 11 Aug 2008 Abstract We study Coxeter racks over Z n and the knot and link invariants they define. We exploit
More informationSelf-distributive quasigroups and quandles
Self-distributive quasigroups and quandles David Stanovský Charles University, Prague, Czech Republic Kazakh-British Technical University, Almaty, Kazakhstan stanovsk@karlin.mff.cuni.cz June 2015 David
More informationarxiv: v1 [math.gr] 18 Nov 2011
A CHARACTERIZATION OF ADEQUATE SEMIGROUPS BY FORBIDDEN SUBSEMIGROUPS JOÃO ARAÚJO, MICHAEL KINYON, AND ANTÓNIO MALHEIRO arxiv:1111.4512v1 [math.gr] 18 Nov 2011 Abstract. A semigroup is amiable if there
More informationTeddy Einstein Math 4320
Teddy Einstein Math 4320 HW4 Solutions Problem 1: 2.92 An automorphism of a group G is an isomorphism G G. i. Prove that Aut G is a group under composition. Proof. Let f, g Aut G. Then f g is a bijective
More informationMATH HL OPTION - REVISION SETS, RELATIONS AND GROUPS Compiled by: Christos Nikolaidis
MATH HL OPTION - REVISION SETS, RELATIONS AND GROUPS Compiled by: Christos Nikolaidis PART B: GROUPS GROUPS 1. ab The binary operation a * b is defined by a * b = a+ b +. (a) Prove that * is associative.
More informationSEMIRINGS SATISFYING PROPERTIES OF DISTRIBUTIVE TYPE
proceedings of the american mathematical society Volume 82, Number 3, July 1981 SEMIRINGS SATISFYING PROPERTIES OF DISTRIBUTIVE TYPE ARIF KAYA AND M. SATYANARAYANA Abstract. Any distributive lattice admits
More informationKyung Ho Kim, B. Davvaz and Eun Hwan Roh. Received March 5, 2007
Scientiae Mathematicae Japonicae Online, e-2007, 649 656 649 ON HYPER R-SUBGROUPS OF HYPERNEAR-RINGS Kyung Ho Kim, B. Davvaz and Eun Hwan Roh Received March 5, 2007 Abstract. The study of hypernear-rings
More informationMATH 433 Applied Algebra Lecture 21: Linear codes (continued). Classification of groups.
MATH 433 Applied Algebra Lecture 21: Linear codes (continued). Classification of groups. Binary codes Let us assume that a message to be transmitted is in binary form. That is, it is a word in the alphabet
More informationCourse 2BA1: Trinity 2006 Section 9: Introduction to Number Theory and Cryptography
Course 2BA1: Trinity 2006 Section 9: Introduction to Number Theory and Cryptography David R. Wilkins Copyright c David R. Wilkins 2006 Contents 9 Introduction to Number Theory and Cryptography 1 9.1 Subgroups
More informationCourse 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra
Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................
More informationClassifying Camina groups: A theorem of Dark and Scoppola
Classifying Camina groups: A theorem of Dark and Scoppola arxiv:0807.0167v5 [math.gr] 28 Sep 2011 Mark L. Lewis Department of Mathematical Sciences, Kent State University Kent, Ohio 44242 E-mail: lewis@math.kent.edu
More information0 Sets and Induction. Sets
0 Sets and Induction Sets A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. We write a A to denote that a is an element of the set
More informationGroups of Prime Power Order with Derived Subgroup of Prime Order
Journal of Algebra 219, 625 657 (1999) Article ID jabr.1998.7909, available online at http://www.idealibrary.com on Groups of Prime Power Order with Derived Subgroup of Prime Order Simon R. Blackburn*
More informationAnalysis of Some Quasigroup Transformations as Boolean Functions
M a t h e m a t i c a B a l k a n i c a New Series Vol. 26, 202, Fasc. 3 4 Analysis of Some Quasigroup Transformations as Boolean Functions Aleksandra Mileva Presented at MASSEE International Conference
More informationBENT POLYNOMIALS OVER FINITE FIELDS
BENT POLYNOMIALS OVER FINITE FIELDS ROBERT S COULTER AND REX W MATTHEWS Abstract. The definition of bent is redefined for any finite field. Our main result is a complete description of the relationship
More informationSYMMETRIC (66,26,10) DESIGNS HAVING F rob 55 AUTOMORPHISM GROUP AS AN. Dean Crnković and Sanja Rukavina Faculty of Philosophy in Rijeka, Croatia
GLASNIK MATEMATIČKI Vol. 35(55)(2000), 271 281 SYMMETRIC (66,26,10) DESIGNS HAVING F rob 55 AUTOMORPHISM GROUP AS AN Dean Crnković and Sanja Rukavina Faculty of Philosophy in Rijeka, Croatia Abstract.
More informationTROPICAL SCHEME THEORY
TROPICAL SCHEME THEORY 5. Commutative algebra over idempotent semirings II Quotients of semirings When we work with rings, a quotient object is specified by an ideal. When dealing with semirings (and lattices),
More informationConjugacy closed loops and their multiplication groups
Journal of Algebra 272 (2004) 838 850 www.elsevier.com/locate/jalgebra Conjugacy closed loops and their multiplication groups Aleš Drápal Department of Mathematics, Charles University, Sokolovská 83, 186
More informationThe Smarandache Bryant Schneider Group Of A Smarandache Loop
The Smarandache Bryant Schneider Group O A Smarandache Loop Tèmító. pé. Gbó. láhàn Jaíyéọlá Department o Mathematics, Obaemi Awolowo University, Ile Ie, Nieria. jaiyeolatemitope@yahoo.com, tjayeola@oauie.edu.n
More informationQuiz 2 Practice Problems
Quiz 2 Practice Problems Math 332, Spring 2010 Isomorphisms and Automorphisms 1. Let C be the group of complex numbers under the operation of addition, and define a function ϕ: C C by ϕ(a + bi) = a bi.
More informationA Generalization of VNL-Rings and P P -Rings
Journal of Mathematical Research with Applications Mar, 2017, Vol 37, No 2, pp 199 208 DOI:103770/jissn:2095-2651201702008 Http://jmredluteducn A Generalization of VNL-Rings and P P -Rings Yueming XIANG
More informationREMARKS 7.6: Let G be a finite group of order n. Then Lagrange's theorem shows that the order of every subgroup of G divides n; equivalently, if k is
FIRST-YEAR GROUP THEORY 7 LAGRANGE'S THEOREM EXAMPLE 7.1: Set G = D 3, where the elements of G are denoted as usual by e, a, a 2, b, ab, a 2 b. Let H be the cyclic subgroup of G generated by b; because
More informationSimple groups and the classification of finite groups
Simple groups and the classification of finite groups 1 Finite groups of small order How can we describe all finite groups? Before we address this question, let s write down a list of all the finite groups
More informationarxiv: v1 [math.ra] 3 Oct 2009
ACTOR OF AN ALTERNATIVE ALGEBRA J.M. CASAS, T. DATUASHVILI, AND M. LADRA arxiv:0910.0550v1 [math.ra] 3 Oct 2009 Abstract. We define a category galt of g-alternative algebras over a field F and present
More informationA Generalization of Boolean Rings
A Generalization of Boolean Rings Adil Yaqub Abstract: A Boolean ring satisfies the identity x 2 = x which, of course, implies the identity x 2 y xy 2 = 0. With this as motivation, we define a subboolean
More informationSpectra of Semidirect Products of Cyclic Groups
Spectra of Semidirect Products of Cyclic Groups Nathan Fox 1 University of Minnesota-Twin Cities Abstract The spectrum of a graph is the set of eigenvalues of its adjacency matrix A group, together with
More informationRELATIVE N-TH NON-COMMUTING GRAPHS OF FINITE GROUPS. Communicated by Ali Reza Ashrafi. 1. Introduction
Bulletin of the Iranian Mathematical Society Vol. 39 No. 4 (2013), pp 663-674. RELATIVE N-TH NON-COMMUTING GRAPHS OF FINITE GROUPS A. ERFANIAN AND B. TOLUE Communicated by Ali Reza Ashrafi Abstract. Suppose
More informationClassification of root systems
Classification of root systems September 8, 2017 1 Introduction These notes are an approximate outline of some of the material to be covered on Thursday, April 9; Tuesday, April 14; and Thursday, April
More informationGroups. 3.1 Definition of a Group. Introduction. Definition 3.1 Group
C H A P T E R t h r e E Groups Introduction Some of the standard topics in elementary group theory are treated in this chapter: subgroups, cyclic groups, isomorphisms, and homomorphisms. In the development
More informationOn completing partial Latin squares with two filled rows and at least two filled columns
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 68(2) (2017), Pages 186 201 On completing partial Latin squares with two filled rows and at least two filled columns Jaromy Kuhl Donald McGinn Department of
More informationCOMMUTATIVE PRESEMIFIELDS AND SEMIFIELDS
COMMUTATIVE PRESEMIFIELDS AND SEMIFIELDS ROBERT S. COULTER AND MARIE HENDERSON Abstract. Strong conditions are derived for when two commutative presemifields are isotopic. It is then shown that any commutative
More informationOn Bruck Loops of 2-power Exponent
On Bruck Loops of 2-power Exponent B.Baumeister, G.Stroth, A.Stein February 12, 2010 Abstract The goal of this paper is two-fold. First we provide the information needed to study Bol, A r or Bruck loops
More informationA DECOMPOSITION OF E 0 -SEMIGROUPS
A DECOMPOSITION OF E 0 -SEMIGROUPS Remus Floricel Abstract. Any E 0 -semigroup of a von Neumann algebra can be uniquely decomposed as the direct sum of an inner E 0 -semigroup and a properly outer E 0
More informationLinear maps. Matthew Macauley. Department of Mathematical Sciences Clemson University Math 8530, Spring 2017
Linear maps Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 8530, Spring 2017 M. Macauley (Clemson) Linear maps Math 8530, Spring 2017
More informationCourse MA2C02, Hilary Term 2013 Section 9: Introduction to Number Theory and Cryptography
Course MA2C02, Hilary Term 2013 Section 9: Introduction to Number Theory and Cryptography David R. Wilkins Copyright c David R. Wilkins 2000 2013 Contents 9 Introduction to Number Theory 63 9.1 Subgroups
More informationFREE STEINER LOOPS. Smile Markovski, Ana Sokolova Faculty of Sciences and Mathematics, Republic of Macedonia
GLASNIK MATEMATIČKI Vol. 36(56)(2001), 85 93 FREE STEINER LOOPS Smile Markovski, Ana Sokolova Faculty of Sciences and Mathematics, Republic of Macedonia Abstract. A Steiner loop, or a sloop, is a groupoid
More informationOn the size of autotopism groups of Latin squares.
On the size of autotopism groups of Latin squares. Douglas Stones and Ian M. Wanless School of Mathematical Sciences Monash University VIC 3800 Australia {douglas.stones, ian.wanless}@sci.monash.edu.au
More informationQuandles and the Towers of Hanoi
The the Bob Laboratory for Algebraic and Symbolic Computation Reem-Kayden Center for Science and Computation Bard College Annandale-on-Hudson, NY 12504 August 8, 2011 The 1 2 3 4 The The Figure Eight (4
More informationFACTOR MAPS BETWEEN TILING DYNAMICAL SYSTEMS KARL PETERSEN. systems which cannot be achieved by working within a nite window. By. 1.
FACTOR MAPS BETWEEN TILING DYNAMICAL SYSTEMS KARL PETERSEN Abstract. We show that there is no Curtis-Hedlund-Lyndon Theorem for factor maps between tiling dynamical systems: there are codes between such
More informationSOME DESIGNS AND CODES FROM L 2 (q) Communicated by Alireza Abdollahi
Transactions on Combinatorics ISSN (print): 2251-8657, ISSN (on-line): 2251-8665 Vol. 3 No. 1 (2014), pp. 15-28. c 2014 University of Isfahan www.combinatorics.ir www.ui.ac.ir SOME DESIGNS AND CODES FROM
More informationDICKSON INVARIANTS HIT BY THE STEENROD SQUARES
DICKSON INVARIANTS HIT BY THE STEENROD SQUARES K F TAN AND KAI XU Abstract Let D 3 be the Dickson invariant ring of F 2 [X 1,X 2,X 3 ]bygl(3, F 2 ) In this paper, we prove each element in D 3 is hit by
More information